SECTION 4.5 SUMMARY

• • The complement $A^C$ of an event A contains all outcomes that are not in A. The union {A or B} of events A and B contains all outcomes in A, in B, and in both A and B. The intersection {A and B} contains all outcomes that are in both A and B, but not outcomes in A alone or B alone.

• • The conditional probability P(B | A) of an event B, given an event A, is defined by

  $$P\left(B\right|A)=\frac{P\left(A\mathrm{and}B\right)}{P\left(A\right)}$$

  when P(A) > 0. In practice, conditional probabilities are most often found from directly available information.

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• • The essential general rules of elementary probability are

  • Legitimate values: 0 ≤ P(A) ≤ 1 for any event A

  • Total probability 1: P(S) = 1

  • Complement rule: P($A^C$) = 1 − P(A)

  • Addition rule: P(A or B) = P(A) + P(B) − P(A and B)

  • Multiplication rule: P(A and B) = P(A)P(B | A)

• • If A and B are disjoint, then P(A and B) = 0. The general addition rule for unions then becomes the special addition rule, P(A or B) = P(A) + P(B).

• • A and B are independent when P(B | A) = P(B). The multiplication rule for intersections then becomes P(A and B) = P(A)P(B).

• • In problems with several stages, draw a tree diagram to organize use of the multiplication and addition rules.